I was watching the Seahawks playoff game on 5 January 2019 and the kicker got injured. For instance, if the Seahawks won that game, could they sign a new player to replace the injured kicker or is the signing period complete after a certain point in the season?
This other answer covers categories of players in which there is a deadline to sign that particular type of player. The answer is correct, but not all-encompassing and (I feel) misses the spirit of your question. For example, during the 2014 season, Ben Tate was signed to the Steelers on December 30th. This is similar to the scenario you are asking about.
Why was Ben Tate signed after a November deadline? Looking at the 2019 NFL Free Agency FAQ:
Q. What determines an unrestricted free agent?
A. A player with four or more accrued seasons whose contract has expired. He is free to sign with any club — with no draft choice compensation owed to his old club — through July 22 or the first scheduled day of the first NFL training camp, whichever is later. His negotiating rights revert exclusively to his old club if by May 7 the old club tendered the player a one-year contract for 110 percent of his prior year’s salary. His old club then has until the Tuesday following Week 10 of the regular season (Nov. 12) to sign him. If he does not sign by that date, he must sit out the season. If no tender is offered by May 7, the player can be signed by any club at any time throughout the season.
Being unable to fully decipher what this says and means and with news outlets rarely getting into contract details in their articles, I conclude that Ben Tate was signed as an unrestricted free agent who didn't have a tender placed on him. Moreover, he was released by the Vikings one week before he was signed by the Steelers on December 23, 2014.
I was watching the Seahawks game on 5 January 2019 during the playoffs and the kicker got injured. If the Seahawks won that game, could they sign a new player to replace him or is the signing period complete after a certain point in the season?
In short, a player in Ben Tate's scenario could be signed. Whether Ben Tate is considered an unrestricted free agent or is considered under a different designation, I am unable to decipher.
I shall note that you do not mention the term "free agent" in your question. To that end, "free agent" criteria would not be helpful if Ben Tate is considered under a different designation.
Upon further research, I found the following source (though it is from 2010) that states:
"Clubs participating in the playoffs are permitted to sign free agents throughout the postseason, but are limited to a total of four free agent signings, including players on other clubs' Practice Squads, during the period that begins at 4 p.m., New York time, on the Wednesday after the final regular season weekend."
Therefore, yes, you may sign a player during the NFL playoffs.
The deadline for extending drafted rookie contracts, franchise player offers, signing a team's unrestricted free agent, and a team's restricted free agent according to NFL.com.
November 13 -- At 4:00 p.m., New York time, signing period ends for Franchise Players who are eligible to receive offer sheets. November 13 -- Prior to 4:00 p.m., New York time, deadline for clubs to sign their unsigned Franchise and Transition Players, including Franchise Players who were eligible to receive offer sheets until this date. If still unsigned after this date, such players are prohibited from playing in the NFL in 2018. November 13 -- Prior to 4:00 p.m., New York time, deadline for clubs to sign their Unrestricted Free Agents to whom the "May 9 Tender" was made. If still unsigned after this date, such players are prohibited from playing in the NFL in 2018. November 13 -- Prior to 4:00 p.m., New York time, deadline for clubs to sign their Restricted Free Agents, including those to whom the "June 1 Tender" was made. If such players remain unsigned after this date, they are prohibited from playing in the NFL in 2018. November 13 -- Prior to 4:00 p.m., New York time, deadline for clubs to sign their drafted rookies. If such players remain unsigned after this date, they are prohibited from playing in the NFL in 2018.